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 Electronic Communications in Probability > Vol. 3 (1998) > Paper 7 open journal systems 


Percolation Dimension of Brownian Motion in R^3

Chad Fargason, Duke University


Abstract
Let $B(t)$ be a Brownian motion in $R^3$. A {it subpath} of the Brownian path $B[0,1]$ is a continuous curve $gamma(t)$, where $gamma[0,1] subseteq B[0,1]$ , $gamma(0) = B(0)$, and $gamma(1) = B(1)$. It is well-known that any subset $S$ of a Brownian path must have Hausdorff dimension $hdim (S) leq 2.$ This paper proves that with probability one there exist subpaths of $B[0,1]$ with Hausdorff dimension strictly less than 2. Thus the percolation dimension of Brownian motion in $R^3$ is strictly less than 2.


Full text: PDF

Pages: 51-63

Published on: February 27, 1998
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Electronic Communications in Probability. ISSN: 1083-589X